Vogel's Universality and its Applications

TL;DR

This thesis explores Vogel's universal formulae for simple Lie algebras, revealing new properties and applying them to formulate refined Chern-Simons theories for all simple gauge groups, including exceptional cases.

Contribution

It introduces new universal formulae in Vogel's framework and applies them to explicitly construct refined Chern-Simons theories for all simple Lie groups.

Findings

New universal formulae in Vogel's description

Additional properties of these formulae discovered

Explicit formulation of refined Chern-Simons theories for all simple gauge groups

Abstract

The present thesis represents developments in two main directions related to the simple Lie algebras. The first one is devoted to the representation theory of the simple Lie algebras. Specifically, we present recent results, which include new universal formulae in Vogel's universal description, as well as the discovery of additional properties of those formulae. In the second part of the thesis, we demonstrate applications of Vogel's description to the study of a physical theory. Namely, we explicitly formulate the { \it refined} Chern-Simons theories on $S^3$ for each of the simple gauge groups, including the exceptional ones.